The RBC approach (Real-Business-Cycle Model) (Romer chapter
5, based on Prescott, 1986, Christiano and Eichenbaum, 1992,
Baxter and King, 1993, Campbell, 1994.
• RBC Objectives: explain business cycles with walrasiens
equilibrium (market clearing conditions), no frictions
• Solution: technology shocks (supply shocks)
• No room for stabilisation policies (monetary and fiscal policy)
• Point of departure: Ramsey model, adding uncertainty and
work/leisure decision in consumer optimization problem.
Yt  K t ( At Lt )1
5.1
K t 1  K t  I t   K t
5.2
 K t  Yt  Ct  Gt   K t
Discret time, G financed by lump-sum taxation, balanced
budget; labour and capital earn their marginal product:
wt  (1   ) K t ( At Lt )  At

 Kt 
 (1   ) 
 At
 At Lt 
5.3
1
 At Lt 
rt   

 Kt 

5.4
The representative household maximises the expected value of:

Nt
U   e u (ct ,1  t )
5.5
H
t 0
1
 t
e 
fn 8 page 181
t
(1   )
 t
ln N t  N  nt ,
n
ut  ln ct  b ln(1  t ),
5.6
b0
Technology and government purchases are the two key variables:
ln At  A  gt  At
5.8
Where A tild reflects the effects of the shocks. We suppose that
A tild follows a first-order autoregressive process:
At   A At 1   A,t
1  A  1
5.9
Where the ε are white-noise disturbances (zero-mean shocks,
uncorrelated with one another). For A, ρ is generally > 0, discussion,
effect of technology shocks gradually disappear through time.
ln Gt  G  ( n  g )t  Gt
5.10
Gt   G Gt 1   G ,t ,  1   G  1
5.11
Where the εG are not correlated with the εA and are white-noises
disturbances that can be viewed as demand (real) shocks.
Household behaviour: intertemporal substitution of labour supply
2 differences with Ramsey: 1) intertemporal substitution of LS and 2) uncertainty
• Consider first the case of one household that live for two periods, without
initial wealth, no uncertainty. The household’s budget constraint is:
1
1
c1 
c2  w1 1 
w2
1 r
1 r
The Lagrangian is:
 ln c1  b ln(1  1 )  e
1

  w1 1 
w2
1 r


5.16
2
ln c2  b ln(1 
1

c2 
2  c1 
1 r 
The control variables are:
c1 , c2 , 1 ,
2
2
)
5.17
The first order conditions for
b
1
  w1
1
and
2
are:
5.18
1
e  b
1

 w2
1 2 1 r
5.19
After manipulation (to demonstrate) we get:
1
1
1
2
1
w2
 
e (1  r ) w1
5.21
Lucas and Rapping, 1969
 w1 (or an  r )  (1  1 )  1  Relative labour supply respond
to relative wages and to r
2 - Optimisation under uncertainty, consider at time t, a marginal
reduction of c that is used to increase wealth and increase
consumption next period in such a way that the household is let
on the optimal path.

U  e
t 0
 t
Nt
(ln ct  b(1  t ))
H
U
 t Nt 1
e
ct
H ct
U
  t N t c
then:
c  e
ct
H ct
This term is on the left-hand side of 4.22, for the right-hand side:
↑c per member in t+1 is:
n
e (1  rt 1 )c
and:
U
  ( t 1) N t 1 1
e
ct 1
H ct 1
e
 t
   ( t 1) N t 1  n 1

N t c
 Et  e
e
(1  rt 1 )  c
H ct
H
ct 1


Demonstrate that this simplifies to 4.23
5.22
 1

1

 e Et 
(1  rt 1 ) 
ct
 ct 1

5.23
Without uncertainty, this is the Euler equation:
ct 1
 e   1  rt 1 
ct
 ct 1  c
ln 
 r
 ct  c
MRS 
U m ,ct
U m ,ct 1
 e   (1  rt 1 )
With uncertainty however, important to note the expectation E:
E ( AB)  E ( A) E ( B)  Cov( A, B)
 1 
 1

1

e
Et 
Et 1  rt 1   Cov 
,1  rt 1 

ct
 ct 1 
 ct 1

5.24
If r(t+1) is high when c(t+1) is also high, then Cov < 0
and this makes saving less attractive since the return to saving is high
when the marginal utility of consumption is low.
And the intratemporal trade-off condition between consumption and
labour supply is (to demonstrate) :
ct
1
t
wt

b
4.26
Analysis of a special case of the model (Long and Plosser, 1983)
-Eliminate government
-Assume 100 percent depreciation
It can be show that in this case, both the saving rate s and labour
supply ℓ are constant (Romer section 4.5, first part).
We concentrate in the analysis that follows. The model resumes to:
ˆ t 1, Lt  ˆ Nt .
Yt  Kt ( At Lt )1 , Kt  sY
ln Yt   ln Kt  (1   )(ln At  ln Lt ).
ln Yt   ln sˆ   ln Yt 1  (1   )(ln At  ln ˆ  ln Nt )
ln Yt   ln sˆ   ln Yt 1  (1   )( A  gt )
 (1   ) A`t  (1   )(ln ˆ  N  nt ) 5.39
The 2 components of the right-hand side
that are not deterministic are:
 ln Yt 1 and (1   ) At 1. Consider Yt  ln Yt  Ytrend
Yt  Yt 1  (1   ) At
5.40
Which implies:
Yt 1  Yt 2  (1   ) At 1
And given that:
5.41'
At   A At 1   A,t
After manipulation (see Romer (2012) p. 205, we then get the following
expression for the departure of the log of output from its normal path:
Yt  (   A )Yt 1   AYt 2  (1   ) A,t
5.42
It follows a second order autoregressive process.
Consider α = 1/3 and ρ = 0.9, we get:
Yt  1.23Yt 1  .33Yt  2   t
Consider the effect of a one time shock of 1/(1-α) to ε(t). The time path
of the effect of the shock on output is: 1, 1.23, 1.22, 1.14. 1.03, 0.94,…
(see analysis on top of page 206)
However, for ρ close to 0.9, the model yields interesting output dynamics
(hump-shaped) (Blanchard, 1981).
Method: Least Squares
Date: 06/04/13 Time: 09:34
Sample: 1955Q1 2013Q1
Included observations: 233
Convergence achieved after 5 iterations
Variable
Coefficient
Std. Error
t-Statistic
Prob.
C
11.25704
1.348679
8.346717
0.0000
AR(1)
1.326158
0.062107
21.35295
0.0000
AR(2)
-0.328169
0.061924
-5.299533
0.0000
R-squared
0.999741
Mean dependent var
8.758554
Adjusted R-squared
0.999738
S.D. dependent var
0.528138
S.E. of regression
0.008541
Akaike info criterion
-6.675036
Sum squared resid
0.016779
Schwarz criterion
-6.630602
Log likelihood
780.6417
Hannan-Quinn criter.
-6.657118
F-statistic
443405.1
Durbin-Watson stat
2.067529
Prob(F-statistic)
0.000000
Inverted AR Roots
1.00
.33
Calibrating (Romer 2012 section 5.8)
Objections to RBC models
• Omission of monetary disturbances (empirically monetary disturbances
affect output) solution : Lucas asymmetric information or incomplete
nominal adjustments à la new-keynesian
• Technology shocks and the Solow residual : the large swings in the Solow
residual from between quarters does not appear to be related with
technology shocks (for the US, Hall 1988 shows that the SR is determined
by the political party of the president, change in military expenditures, and
oil price)
• Intertemporal substitution in labour supply
• Dynamics of the basic RBC does not look like a business cycle (unless a
precise dynamics is assumed for the shocks).
Extensions of RBC models
• ‘Real extension’ : indivisible labor (Rogertson and
Hansen), labor supply is either 0 or 1. Make the model
labor supply more sensible to wage. Increase the SD of
output in 5.4 from 1.3 to 1.73.
• Distortionary taxes and many others (see footnote 33
page 231)
• Introducing nominal rigidity read page 231 to 233.